Optimization
In regular temperament theory, optimization is the theory and practice to find low-error tunings of regular temperaments.
A regular temperament is defined by a mapping or a comma basis. It does not contain specific tuning information. To tune a temperament, one must define a tuning map by specifying the size of each generator. The question is what it should be. In general, a temperament is an approximation to just intonation (JI). Any tuning will unavoidably introduce errors on some intervals for sure. The art of tempering seems to be about compromises – to find a sweet spot where the concerning intervals have the least overall error, so that the harmonic qualities of JI are best preserved.
Taxonomy
Roughly speaking, there are two types of tunings with diverging philosophies: prime-based tunings and target tunings.
- A prime-based tuning is optimized for the formal primes, but they are representative for the set of all intervals. There are two equivalent perspectives. First, in the tuning space, it minimizes the errors of formal primes. Second, in the interval space, it rates the complexity of all intervals through a norm, and it minimizes the maximum damage (i.e. error divided by complexity) for all intervals.
- A target tuning is optimized for a particular set of intervals and considers the rest irrelevant. However, the interval does not get infinite complexity even if it is disregarded due to the normed nature of the interval space, so these tunings also correspond to all-interval damage minimizations of some sorts.
This article focuses on prime-based tunings. See the dedicated page (→ Target tunings) for target tunings.
Norm
In order to perform prime-based optimization, all intervals must be rated by complexity, so it is critical to employ a norm. Technically, this is to embed the just intonation group into a normed vector space. There are a few aspects to consider. The weight, which determines how important each formal prime is, and the skew, which determines how divisive ratios are more important than multiplicative ratios. They can be interpreted as transformations of either the norm or the coordinates of the space. The two views are equivalent. In addition, there is the order (or sometimes just dubbed the norm), which determines how the space can be traversed.
Weight
The weight, represented by a diagonal transformation matrix, determines the importance of each formal prime. Since the tuning space and the interval space are dual to each other, rating of importance in the tuning space is equivalent to rating of complexity in the interval space. The Tenney weight is the most common weight:
[math]\displaystyle W = \operatorname {diag} (1/\log_2 (Q)) [/math]
which indicates that the prime harmonic q in Q = ⟨2 3 5 …] has the importance of 1/log_{2}(q). Its dual states that q has the complexity of log_{2}(q).
Skew
An orthogonal space treats divisive ratios as equally important as multiplicative ratios, yet divisive ratios are sometimes thought to be more important. For example, 5/3 is sometimes found to be more important than 15. The skew is introduced to address that.
Notably, adopting the standard Weil height will skew the space to 60 degrees.
Both the weight and the skew are represented by matrices that can be applied to the mapping. In a more general sense, the distinction may not matter, and they may be collectively called weight–skew transformation.
Order
The order of the norm determines what is a unit step in the interval space.
The Euclidean norm aka L^{2} norm resembles real-world distances.
The Manhattan norm or taxicab norm a.k.a. L^{1} norm resembles movement of taxicabs in Manhattan – it can only traverse horizontally or vertically. A diagonal movement counts as two steps.
The Chebyshevian norm a.k.a. L^{inf} norm is the opposite of the Manhattan norm – it is the maximum number of steps along any axis, so a diagonal movement is the same as a horizontal or vertical one.
Note that the dual norm of L^{1} is L^{inf}, and vice versa. Thus, the Manhattan norm corresponds to the L^{inf} tuning space, and the Chebyshevian norm corresponds to the L^{1} tuning space. The dual of L^{2} norm is itself, so the Euclidean norm corresponds to Euclidean tuning as one may expect.
Enforcement
Enforcement is the technique where certain intervals are forced to be tuned as desired. The octave is the most common interval subject to enforcement, and is often assumed to be the enforced interval, but other intervals are possible. The two common methods are destretch and constraint.
Destretch
Destretch is a postprocess to enforce a pure interval. The result is no longer optimal measured by the original norm, but it often works as a quick approximation to more sophisticated tunings. The most common destretched tuning is POTE tuning, which approximates CTWE tuning.
Constraint
Constraint is a logical method to enforce one or more pure intervals. A pure interval added this way is known as an eigenmonzo a.k.a. unchanged-interval. It defines a feasible region for optimization, and the result measured by the original norm is feasibly optimal.
General formulation
In general, the temperament optimization problem (except for the destretch) can be defined as follows. Given a temperament mapping V and the just tuning map J, we specify a weight–skew transformation, represented by transformation matrix X, and a q-norm. An optional eigenmonzo list M_{C} can be added. The goal is to find the generator list G by
Minimize
[math]\displaystyle \lVert GV_X - J_X \rVert_q [/math]
subject to
[math]\displaystyle (GV - J)M_{\rm C} = O [/math]
where (·)_{X} denotes the variable in the weight–skew transformed space, found by
[math]\displaystyle \begin{align} V_X &= VX \\ J_X &= JX \end{align} [/math]
Common tunings
A good number of common tuning schemes have been given names. The following table shows some of them by weight–skew against the order.
Weight–Skew\Order | Chebyshevian (L^{1} tuning) |
Euclidean (L^{2} tuning) |
Manhattan (L^{infinity} tuning) |
---|---|---|---|
Tenney Tenney-Weil |
TC tuning |
TE tuning TWE tuning |
TOP tuning |
Equilateral Equilateral-Weil |
EC tuning |
Frobenius tuning EWE tuning |
EOP tuning |
Benedetti/Wilson Benedetti/Wilson-Weil |
BC tuning |
BE tuning BWE tuning |
BOP tuning |
Each has a constrained and/or destretched variant. E.g. for TE tuning there is CTE tuning, and for TWE tuning there is CTWE tuning.
See also
- A Middle Path between Just Intonation and the Equal Temperaments – Part 1 ("middle path") by Paul Erlich
External links
- Prime Based Error and Complexity Measures ("primerr.pdf") by Graham Breed